A Fuzzy Expectation Maximization based
method for estimating the parameters of a
multi-state degradation model from...
Article in Annals of Nuclear Energy · July 2017 DOI: 10.1016/j.anucene.2017.07.017 CITATIONS 0 READS 60 4 authors, including: Some of the authors of this publication are also working on these related projects: Risk and Safety Management View project Advanced computational methods for modelling the mechanisms of degradation in equipments of electricity production plants and uncertainty modelling and propagation View project Francesco Cannarile Politecnico di Milano 6 PUBLICATIONS 6 CITATIONS SEE PROFILE Michele Compare Politecnico di Milano 48 PUBLICATIONS 241 CITATIONS SEE PROFILE Enrico Zio Politecnico di Milano 827 PUBLICATIONS 9,184 CITATIONS SEE PROFILEAll content following this page was uploaded by Francesco Cannarile on 28 July 2017. The user has requested enhancement of the downloaded file.
A Fuzzy Expectation Maximization based method for estimating the
parameters of a multi-state degradation model from imprecise
maintenance outcomes
F. Cannarile1,2, M. Compare1,2, E. Rossi1, E. Zio1,2,3
1Dipartimento di Energia, Politecnico di Milano, Italy 2Aramis s.r.l., Milano, Italy
3Chair on Systems Science and Energetic Challenge, Foundation Electricité de France at Ecole
CentraleSupelec, France
Abstract
Multi-State (MS) reliability models are used in practice to describe the evolution of degradation in industrial components and systems. To estimate the MS model parameters, we propose a method based on the Fuzzy Expectation-Maximization (FEM) algorithm, which integrates the evidence of the field inspection outcomes with information taken from the maintenance operators about the transition times from one state to another. Possibility distributions are used to describe the imprecision in the expert statements. A procedure for estimating the Remaining Useful Life (RUL) based on the MS model and conditional on such imprecise evidence is, then, developed. The proposed method is applied to a case study concerning the degradation of pipe welds in the coolant system of a Nuclear Power Plant (NPP). The obtained results show that the combination of field data with expert knowledge can allow reducing the uncertainty in degradation estimation and RUL prediction.
Key words: Multi-State Systems, Homogeneous Continuous-Time Semi-Markov Process (HCTSMP), Weibull Distribution, Fuzzy Expectation-Maximization (FEM), Residual Useful Life (RUL), Piping System (PS), Nuclear Power Plants (NPPs).
Acronyms and symbols
CDF Cumulative Distribution Function EM Expectation-Maximization FEM Fuzzy Expectation-Maximization
HCTSMM Homogeneous Continuous-Time Semi-Markov Model MC Monte Carlo
MS Multi-State
MLE Maximum Likelihood Estimation NPP Nuclear Power Plant
PDF Probability Density Function PFM Probabilistic Fracture Mechanics PS Piping System
PWR Pressurized Water Reactor RCS Reactor Cooling System RUL Remaining Useful Life 𝐶0 Case 0
𝐶1 Case 1: moderately risk-averse expert 𝐶2 Case 2: risk averse expert
𝐶3 Case 3: risk prone expert 𝐷 Dataset of inspection outcomes
𝐸 State space
𝑓𝑇𝑖→𝑖+1 PDF of 𝑇𝑖→𝑖+1
𝑓𝑇𝑖→𝑖+1(∙ |𝑡𝑛,𝑖0 ) Conditional PDF of 𝑇𝑖→𝑖+1 provided that 𝑇𝑖→𝑖+1≥ 𝑡𝑛,𝑖0
𝑓𝑅𝑈𝐿(𝑘𝜏) PDF of 𝑅𝑈𝐿(𝑘𝜏)
𝐹𝑇𝑖→𝑖+1 CDF of 𝑇𝑖→𝑖+1
𝐹𝑓𝑎𝑖𝑙𝑢𝑟𝑒 CDF of 𝑇𝑓𝑎𝑖𝑙𝑢𝑟𝑒
𝑓̃𝑇𝑖→𝑖+1 PDF of fuzzy observations
𝑘𝑛,𝑖 Inspection at which the 𝑛𝑡ℎcomponent is found in state 𝑖 for the first time
𝐿 Likelihood function
𝐿̃ Likelihood function of fuzzy observations
ℒ𝑖→𝑖+1 𝑖𝑡ℎ contribution to the log-likelihood function
ℒ̃𝑖→𝑖+1 𝑖𝑡ℎ contribution to log-likelihood function of fuzzy observations
ℒ𝑙𝑜𝑔 Log-likelihood function
ℒ̃𝑙𝑜𝑔 Log-likelihood function of fuzzy observations
𝑀𝑛 Number of inspections on component 𝑛
𝑁 Number of components
𝑄 Log-likelihood function conditional on fuzzy evidence
𝑅𝑇𝑖→𝑖+1 Reliability function of 𝑇𝑖→𝑖+1
𝑅𝑓𝑎𝑖𝑙𝑢𝑟𝑒 Reliability function of 𝑇𝑓𝑎𝑖𝑙𝑢𝑟𝑒
𝑡 Time
𝑡𝑛 Vector of transition times of the 𝑛𝑡ℎ component
𝑇𝑖→𝑖+1 Transition time from state 𝑖 to state 𝑖 + 1, random variable
𝑇𝑚 Mission time
𝑇𝑓𝑎𝑖𝑙𝑢𝑟𝑒 Failure time
𝑡𝑛,𝑖→𝑖+1 Transition time of the 𝑛𝑡ℎ component from state 𝑖 to state 𝑖 + 1, observed value
𝑡̃𝑛,𝑖→𝑖+1 Fuzzy transition time, observed value
𝑡𝑛,𝑖→𝑖+1 Lower bound of the support of 𝜇𝑡̃𝑛,𝑖→𝑖+1(𝑡𝑖→𝑖+1)
𝑡⏞𝑛,𝑖→𝑖+1 Core of 𝜇𝑡̃𝑛,𝑖→𝑖+1(𝑡𝑖→𝑖+1)
𝑡𝑛,𝑖→𝑖+1 Upper bound of the support of 𝜇𝑡̃𝑛,𝑖→𝑖+1(𝑡𝑖→𝑖+1)
𝑡𝑛,𝑖0 Sojourn time in state 𝑖 of the 𝑛𝑡ℎ component
𝑡̇𝑛,𝑖0 Elapsed time from the first inspection time in which the component has been found in state 𝑖, and the last one
𝛼𝑖 Scale parameter of the Weibull distribution describing the uncertainty on the
transition time from state 𝑖 to state 𝑖 + 1
𝛽𝑖 Shape parameter of the Weibull distribution describing the uncertainty on the
transition time from state 𝑖 to state 𝑖 + 1 𝜹 Vector of 𝛿𝑛, 𝑛 = 1 … 𝑁
𝛿𝑛 Vector of binary variables associated to the 𝑛𝑡ℎcomponent
𝛿𝑛,𝑖→𝑖+1 Binary variable associated to the 𝑛𝑡ℎcomponent indicating the censoring of the
transition time from state 𝑖 to state 𝑖 + 1
𝜆𝑖→𝑖+1 Transition rate from state 𝑖 to state 𝑖 + 1
𝜆𝐶0 Transition rate for Case 𝐶0
𝜆𝐶2 Transition rate for Case 𝐶2
𝜆𝐶3 Transition rate for Case 𝐶3 𝜇𝑡̃𝑛,𝑖→𝑖+1 Possibility distribution on 𝑡̃𝑛,𝑖→𝑖+1
𝜏 Interval between two successive inspections 𝜗 Vector of the transition time parameters vector 𝜗̂𝑚𝑙𝑒 MLE estimates of 𝜗
𝜗𝑞 Estimates of 𝜗 at iteration 𝑞 𝑖 state index, 𝑖 = 1,2,3
𝑘 inspection index, 𝑘 = 1 … 𝑀𝑛 𝑛 component index, 𝑛 = 1. . 𝑁 𝑞 FEM iteration index
1. INTRODUCTION
Multi-State (MS) degradation modelling is receiving considerable attention in the domain of reliability and maintenance engineering (Zio, 2016), due the fact that MS models offer a description
of the degradation evolution which is more realistic than that given by binary models: the evolution of many degradation processes proceeds in successive phases, which reflect the relative degree of deterioration (Moghaddass & Zuo, 2014). A further reason which justifies the growing interest in MS degradation models is their fit with the field maintenance data acquired from the operating systems. For example, operators typically assign a qualitative tag to the equipment health during periodic inspections such as ‘not degraded’, ‘slightly degraded’, ‘badly degraded’, etc.
Given these characteristics, MS models have been adopted to describe the evolution of degradation of components of diverse application fields: membranes of pumps operating in Nuclear Power Plants (NPPs) (Baraldi et al., 2011), turbine nozzles for the Oil&Gas industry (Compare et al., 2016), turbofan engines (Moghaddas & Zuo, 2014), Diesel engines (Giorgio et al., 2011), to cite a few. A Multi-State (MS) degradation model has also been developed in (Fleming and Smit, 2008) for the Piping System (PS) of NPPs, where PSs are highly risk-sensitive structural elements (Gopika et al., 2003; Di Maio et al., 2015). In details, in the model by (Fleming & Smit, 2008), which is general enough to represent all known NPP pipe failure mechanisms (Fleming, 2004), the degradation process affecting a PS is discretized into four states, each one associated to a physically different phenomenon, with state transition rates that are taken constant over time and, consequently, sojourn times in each state that obey exponential distributions (e.g., Fleming & Smit, 2008). However, it has been shown in (Veeramany & Pandey, 2011; Chatterjee & Modarres, 2008), that the constant rate assumption is not coherent with the evidence coming from many real industrial applications. Thus, to overcome the limitation of constant transition rates, the theoretical framework of the Homogeneous Continuous-Time Semi-Markov Processes (HCTSMPs, Howard, 1964) has been embraced to develop MS degradation models, which allow considering arbitrary sojourn time distributions, thus, taking into account the influence of the history of the degradation process on its future evolution. In particular, (Veeramany & Pandey, 2011) developed a HCTSMP model to describe the degradation of PSs in NPPs.
For practical application, the estimation of the parameters of the MS semi-Markov degradation model, with associated uncertainty, is fundamental and different approaches have been proposed in the literature to adjust the model to the knowledge, information and data available.
When sufficient field data is available, statistical techniques such as Maximum Likelihood Estimation (MLE) can be adopted (Zio, 2007; Gosselin & Fleming, 1997). However, the availability of rich datasets of NPP PS degradation and maintenance data is not typical and the problem of parameter estimation is further complicated by at least two other aspects:
• The inherent complexity of the PSs in NPPs and diversity in the degradation influenced by operating and ambient conditions (Tipping, 2010); then, it becomes difficult to identify mechanisms and homogeneous populations of PS for statistical inference.
• The possible noninformativeness of the data, i.e., of the outcomes of inspections performed every 2-5 years, in which the PS is typically found in the first degradation states, due to its very high reliability (Nánási, T., 2014; Fleming, 2004; Veeramany & Pandey, 2011; Simonen & Goselin, 2001).
With this scarcity of data, it is necessary to exploit any additional knowledge or information available to build more accurate reliability models (Zio, 2016). In this respect, Probabilistic Fracture Mechanics (PFM) models have been developed to predict PS crack initiation and growth from existing flaws (Verma & Srividya, 2011), which combine the knowledge about the physics of the crack propagation, modelled as a stochastic process, with PS service data that are used to tune the PFM model parameters. However, (Fleming, 2004) pointed out that one main limitation of the PFM approach is that the data used for model setting reflect the influence of previous PS inspection programs; thus, changes in these programs may introduce biases in the transition rates estimates.
In the present work, we consider that additional information on the occurrence of state transitions can be obtained from experts to supplement field data. Namely, we assume that experts can give statements such as “The pipe transition from detectable flaw state to detectable leak state occurred between 1998 and 2000, March 1999 being the expected month for this transition”. Obviously, the imprecision in these qualitative statements need to be properly represented and combined with the field data.
Bayesian statistics is often adopted to this aim, starting from the elicitation from experts of prior distributions of the model parameters and following with their update based on to the field evidence collected (Compare et al., 2017a). Markov Chain Monte Carlo algorithms (Robert & Casella, 2004) can be used to estimate the posterior distributions of the multi-state model parameters, which encode both the prior expert knowledge and the field evidence. However, the representation by probability distributions of the imprecision in the qualitative expert statements is debatable, as it has been argued that the probabilistic approach in situations of scarce evidence tends to force assumptions that may not be justified by the available information (Aven et al., 2014; Baudrit et al., 2008; Bowles & Peláez, 1995). For this reason, we here use possibility distributions to represent the imprecision in the expert statements about the transition times, to “imperfectly specify a value that is existing and precise, but not measurable with exactitude under the given observation conditions” (Denoeux, 2011).
To estimate the MS model parameters from partially observed data, we resort to the Fuzzy Expectation-Maximization (FEM) algorithm (Denoeux, 2011); this uses the Zadeh’s extension principle to extend the application of standard statistical approaches to possibility distributions, which are formally coincident with the membership functions of fuzzy sets (Dubois, 2006).
Finally, based on the MS degradation model, we propose a methodology to estimate the Remaining Useful Life (RUL) of the NPP PSs.
To sum up, the main contributions of the present work are:
1. The development of a methodology to estimate the parameters of a MS degradation model, which exploits both data from inspection outcomes and expert information.
2. The development of a methodology to estimate the RUL, conditional on imprecise evidence. The remainder of the paper is organized as follows. Section 2 describes the problem settings, the available information and data. Section 3 illustrates the methodology to estimate the unknown parameters of the HCTSMP model in the considered settings. In Section 4, the methodology to estimate the RUL is illustrated. The application of the developed methodologies to a case study concerning simulated PS degradation paths is reported in Section 5. Finally, Section 6 concludes the work.
2. Problem Settings
To focus concretely the illustration of our work, we develop a MS model derived from the 4-states model proposed by (Fleming, 2004) for describing degradation in PSs of NPPs (Figure 1).
Figure 1: Sketch of MS model.
In state 1, the PS is assumed to be in an as good as new state; flaws are present but not detectable. These gradually grow until they become detectable, whose condition is represented by state 2. Then, the PS further degrades and a leak becomes detectable (state 3). Finally, the leak extends until it leads to rupture (state 4) (Veeramany & Pandey, 2011). Pipes are assumed to be non-repairable: this means
that in the representation of the model (Figure 1), the transitions only go from left-to-right and also that state 4 is an absorbing state (i.e., once reached, it cannot be left).
The random transition time, 𝑇𝑖→𝑖+1, from state 𝑖 to state 𝑖 + 1 is assumed to obey a Weibull
distribution (Cannarile et al., 2015a), with scale parameter 𝛼𝑖 and shape parameter 𝛽𝑖, 𝑖 = 1, 2, 3. The probability density function (PDF) of 𝑇𝑖→𝑖+1 is given by
𝑓𝑇𝑖→𝑖+1(𝑡𝑖→𝑖+1) =𝛽𝑖 𝛼𝑖∙ ( 𝑡𝑖→𝑖+1 𝛼𝑖 ) (𝛽𝑖−1) exp (− (𝑡𝑖→𝑖+1 𝛼𝑖 ) 𝛽𝑖 ) 𝑡𝑖→𝑖+1 > 0, 𝛼𝑖 > 0, 𝛽𝑖 > 0 (1)
and the corresponding Cumulative Distribution Function (CDF) 𝐹𝑇𝑖→𝑖+1(𝑡𝑖→𝑖+1), reliability function
𝑅𝑇𝑖→𝑖+1(𝑡𝑖→𝑖+1) and transition rate 𝜆𝑇𝑖→𝑖+1(𝑡𝑖→𝑖+1) are, respectively:
𝐹𝑇𝑖→𝑖+1(𝑡𝑖→𝑖+1) = ∫ 𝑓𝑇𝑖→𝑖+1(𝑡) 𝑡𝑖→𝑖+1 0 𝑑𝑡 = 1 − 𝑒 −(𝑡𝑖→𝑖+1𝛼 )𝛽𝑖 (2) 𝑅𝑇𝑖→𝑖+1(𝑡𝑖→𝑖+1) = exp (− ( 𝑡𝑖→𝑖+1 𝛼𝑖 ) 𝛽𝑖 ) 𝑡𝑖→𝑖+1> 0, 𝛼𝑖 > 0, 𝛽𝑖 > 0 (3) 𝜆𝑇𝑖→𝑖+1(𝑡𝑖→𝑖+1) = 𝛽𝑖 𝛼𝑖∙ ( 𝑡𝑖→𝑖+1 𝛼𝑖 ) (𝛽𝑖−1) 𝑡𝑖→𝑖+1 > 0, 𝛼𝑖 > 0, 𝛽𝑖 > 0 (4)
The choice of relying on Weibull distributions is justified by practical reasons: Weibull distributions are the probability distributions most commonly used in reliability engineering to describe the degradation processes of industrial components in the semi-Markov framework (Boutros et al., 2011) (Moghadass & Zuo, 2012), (Compare et al. 2017b), (Giorgio et al., 2011) due to their flexibility and the clear meaning of the distribution parameters. For this reason, experts of different industrial fields feel comfortable with using Weibull distributions to characterize the evolution of the degradation processes (Cannarile et al., 2017).
2.1.
Available data
We assume that a dataset 𝐷 is available containing the inspection outcomes of 𝑁 NPP PSs, whose degradation evolves according to the HCTSMM described above. We also assume that each
component is perfectly working (i.e., it is in state 1) at time 𝑡 = 0 and is inspected with period 𝜏 over the mission time 𝑇𝑚.
We indicate by 𝑀𝑛 the number of inspections performed on the 𝑛𝑡ℎ component through its mission
time 𝑇𝑚, whereas 𝑘𝑛,𝑖 represents the first inspection at which the 𝑛𝑡ℎ component is found in state 𝑖,
with 𝑘𝑛,1 = 0 and 𝑘𝑛,𝑖 ∈ {1, … , 𝑀𝑛}, 𝑖 = 2,3,4.
In this setting, the transition time 𝑡𝑛,𝑖→𝑖+1 of the 𝑛𝑡ℎ PS, 𝑛 = 1 … 𝑁, from state 𝑖 to state 𝑖 + 1, 𝑖 =
1, 2, 3, can be regarded as a realization of the random variable 𝑇𝑖→𝑖+1, induced by randomly sampling from a population of NPP PSs (Denoeux, 2011), although a censoring mechanism avoids observing transition times that would occur after the time horizon 𝑇𝑚. In this respect, if 𝑖𝑛∗ =
max
𝑖∈{1,2,3,4}(∑ 𝑡𝑛,𝑗−1→𝑗 < 𝑇𝑚
𝑖
𝑗=1 ) < 4, where 𝑡𝑛,0→1 = 0, then for simplicity we set 𝑘𝑛,𝑠+1 = 𝑀𝑛, ∀ 𝑠 ≥
𝑖∗∩ 𝑠 ∈ {2,3,4}. For example, if the 𝑛𝑡ℎ component is found in state 3 at the end of the mission time 𝑇𝑚, then 𝑖∗ = 3, 𝑘𝑛,3= 𝑘𝑛,4= 𝑀𝑛, whereas 𝑡𝑛,3→4 is unknown, but larger than 𝑇𝑚− (𝑡𝑛,1→2+
𝑡𝑛,2→3).
On this basis, we introduce the binary variable for 𝑖 = 1, 2, 3
𝛿𝑛,𝑖→𝑖+1= {0 𝑖𝑓 ∑ 𝑡𝑛,𝑗−1→𝑗 𝑖 𝑗=1 < 𝑇𝑚 𝑎𝑛𝑑 𝑡𝑛,𝑖→𝑖+1≤ [𝑇𝑚− ∑ 𝑡𝑛,𝑗−1→𝑗 𝑖 𝑗=1 ] 1 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (5)
to indicate whether 𝑡𝑛,𝑖→𝑖+1 is an actual transition time or a right-censored observation (Zio, 2007). In words, 𝛿𝑛,𝑖→𝑖+1 is set to 0 if the transition from state 𝑖 to state 𝑖 + 1 occurred before 𝑇𝑚, and to 1 otherwise (Cannarile et al., 2015b).
Moreover, even for uncensored transitions, 𝑡𝑛,𝑖→𝑖+1 cannot be directly observed; rather, we know that
(𝑘𝑛,𝑖+1− 1)𝜏 ≤ [ ∑𝑖𝑗=1𝑡𝑛,𝑗−1→𝑗+ 𝑡𝑛,𝑖→𝑖+1] ≤ 𝑘𝑛,𝑖+1𝜏 , 𝑘 = 1, … , 𝑀𝑛 and 𝑖 = 1, 2, 3.
Formally, the available dataset can be represented by 𝐷 = (𝒕, 𝜹), where 𝒕 = [𝑡1, … 𝑡𝑁], 𝑡𝑛 = [𝑡𝑛,0→1, … , 𝑡𝑛,𝑖𝑛∗−1→𝑖
𝑛
∗], 𝜹 = [𝛿1, … , 𝛿𝑁], and 𝛿𝑛 = [𝛿𝑛,1→2, 𝛿𝑛,2→3, 𝛿𝑛,3→4], 𝑛 = 1, … , 𝑁.
2.2.
Information from experts
When the expert inspects the component, he/she can add additional information on the transition times, based on his/her knowledge. We assume that expert statements about the unknown 𝑡𝑛,𝑖→𝑖+1
1. An interval [𝑡𝑛,𝑖→𝑖+1, 𝑡𝑛,𝑖→𝑖+1] in which the transition certainly occurred, where
[ ∑𝑖𝑗=1𝑡𝑛,𝑗−1→𝑗+ 𝑡𝑛,𝑖→𝑖+1] ≥ (𝑘𝑛,𝑖+1− 1)𝜏; [ ∑𝑖𝑗=1𝑡𝑛,𝑗−1→𝑗+ 𝑡𝑛,𝑖→𝑖+1] ≤ 𝑘𝑛,𝑖+1𝜏.
2. A time instant 𝑡⏞𝑛,𝑖→𝑖+1 ∈ [𝑡𝑛,𝑖→𝑖+1, 𝑡𝑛,𝑖→𝑖+1] in which the transition occurrence is fully plausible.
For simplicity, these pieces of information are represented by triangular possibility distributions:
ℒ𝜇𝑡̃
𝑛,𝑖→𝑖+1(𝑡𝑖→𝑖+1) =(𝑡𝑛,𝑖→𝑖+1, 𝑡⏞𝑛,𝑖→𝑖+1, 𝑡𝑛,𝑖→𝑖+1) (6)
with support [𝑡𝑛,𝑖→𝑖+1, 𝑡𝑛,𝑖→𝑖+1] and core 𝑡⏞𝑛,𝑖→𝑖+1, (see Figure 2). Namely, 𝜇𝑡̃𝑛,𝑖→𝑖+1(𝑡𝑖→𝑖+1) expresses the degree of possibility that the true value of 𝑇𝑖→𝑖+1 is 𝑡𝑖→𝑖+1: when 𝜇𝑡̃𝑛,𝑖→𝑖+1(𝑡𝑖→𝑖+1) = 0, then the
outcome 𝑡𝑖→𝑖+1 is considered impossible by the expert, whereas 𝜇𝑡̃𝑛,𝑖→𝑖+1(𝑡𝑖→𝑖+1) = 1 means that the
outcome 𝑡𝑖→𝑖+1 is fully plausible, expected by the expert (Aven et al., 2014).
Notice that the triangular shape for the possibility distribution is the appropriate choice when the expert is willing to specify the most likely value that 𝑡𝑛,𝑖→𝑖+1 can assume (Aven et al., 2014).
Figure 2: Possibility distribution example for the information 𝒕𝒏,𝒊→𝒊+𝟏= 𝟑, 𝒕⏞𝒏,𝒊→𝒊+𝟏= 𝟑. 𝟔𝟔𝟕, 𝒕̅𝒏,𝒊→𝒊+𝟏= 𝟒.
3. Parameter Estimation
The aim of this Section is estimating the parameters (𝛼𝑖, 𝛽𝑖) of the Weibull distributions. We consider two different situations:
• The only source of information available is dataset 𝐷. In this standard case, we can apply the MLE approach.
• Also information provided by experts about transition times is available, described in the form of possibility distributions 𝜇𝑡̃𝑛,𝑖→𝑖+1(𝑡𝑖→𝑖+1), 𝑖 = 1, 2, 3, 𝑛 = 1, … , 𝑁. In this case, we use the
FEM algorithm proposed in (Denoeux, 2011).
The comparison of these two settings allows highlighting the benefit of exploiting the information coming from experts.
For clarity of presentation, the distribution parameters are indicated by 𝜗 = [𝜗1, 𝜗2, 𝜗3], where 𝜗𝑖 =
[𝛼𝑖, 𝛽𝑖], 𝑖 = 1,2,3.
3.1. Estimation based on inspection outcomes, only
The application of MLE requires defining the likelihood function, which is given by:
𝐿(𝜗|𝐷) = ∏ [∏ [ 𝐹𝑇𝑖→𝑖+1(𝑘𝑛,𝑖+1𝜏) − 𝐹𝑇𝑖→𝑖+1((𝑘𝑛,𝑖+1− 1)𝜏)] 𝛿𝑛,𝑖→𝑖+1 ∙ 𝑅𝑇𝑖→𝑖+1(𝑇𝑚− 𝑘𝑛,𝑖𝜏) (1−𝛿𝑛,𝑖→𝑖+1) 3 𝑖=1 ] 𝑁 𝑛=1 (7)
where the difference 𝐹𝑇𝑖→𝑖+1(𝑘𝑛,𝑖+1𝜏) − 𝐹𝑇𝑖→𝑖+1((𝑘𝑛,𝑖+1− 1)𝜏) represents the probability of finding the component for the first time in state 𝑖 + 1 at inspection 𝑘𝑛,𝑖+1, provided that it was in state 𝑖 at
inspection (𝑘𝑛,𝑖− 1), whereas 𝑅𝑇𝑖→𝑖+1(𝑇𝑚− 𝑘𝑛,𝑖𝜏) indicates the probability of spending time (𝑇𝑚− 𝑘𝑛,𝑖𝜏) in state 𝑖. The quantity 𝛿𝑛,𝑖→𝑖+1 determines which of the two contributions has to be considered for the 𝑛𝑡ℎ component, depending on the censoring mechanism it has undergone.
The corresponding log-likelihood can be written as:
ℒ(𝜗|𝐷) = ℒ1→2(𝜃1|𝐷) + ℒ2→3(𝜃2|𝐷) + ℒ3→4(𝜃3|𝐷) = ∑ ℒ𝑖→𝑖+1(𝜃𝑖|𝐷) 3 𝑖=1 (8) where ℒ𝑖→𝑖+1(𝜗𝑖|𝐷) = ∑ log ([ 𝐹𝑇𝑖→𝑖+1(𝑘𝑛,𝑖+1𝜏) − 𝐹𝑇𝑖→𝑖+1((𝑘𝑛,𝑖+1− 1)𝜏)] 𝛿𝑛,𝑖→𝑖+1 𝑅𝑇𝑖→𝑖+1(𝑇𝑚− 𝑘𝑛,𝑖𝜏) 1−𝛿𝑛,𝑖→𝑖+1 ) 𝑁 𝑛=1 ∑ 𝑡𝑛,𝑗−1→𝑗 𝑖 𝑗=1 < (𝑘𝑛,𝑖− 1)𝜏, 𝑖 = 1,2,3 (9)
𝜗̂𝑚𝑙𝑒 = argmax 𝜗∈𝛩
ℒ(𝜗|𝐷) (10)
Notice that Equations (8) - (9) simplify the estimation of 𝜗̂𝑚𝑙𝑒. In fact, each of the three contributions
ℒ𝑖→𝑖+1, 𝑖 = 1, 2, 3 depends on the two parameters (𝛼𝑖, 𝛽𝑖), only. Then, one can divide the
maximization problem in Equation (7) into three simpler sub-problems, which can be solved independently on each other.
3.2. Estimation based on inspection outcomes and information elicited from experts
To show the methodology to estimate 𝜗 based on the possibility distributions 𝜇𝑡̃𝑛,𝑖→𝑖+1, we first derive
the likelihood function as if we exactly knew the transition times 𝑡𝑛,𝑖→𝑖+1. On this basis, we will easily extend this function to the case of imprecise transition times 𝑡̃𝑛,𝑖→𝑖+1.
Analogously to the previous case, the likelihood function reads:
𝐿(𝜗|𝐷) = ∏ [∏ 𝑓𝑇𝑖→𝑖+1(𝑡𝑛,𝑖→𝑖+1) 𝛿𝑛,𝑖→𝑖+1 ∙ 𝑅𝑇𝑖→𝑖+1(𝑇𝑚− ∑ 𝑡𝑛,𝑗−1→𝑗 𝑖 𝑗=1 ) (1−𝛿𝑛,𝑖→𝑖+1) 3 𝑖=1 ] 𝑁 𝑛=1 (11)
where the pdf 𝑓𝑇𝑖→𝑖+1(𝑡𝑛,𝑖→𝑖+1) is used instead of 𝐹𝑇𝑖→𝑖+1(𝑘𝑛,𝑖+1𝜏) − 𝐹𝑇𝑖→𝑖+1((𝑘𝑛,𝑖+1− 1)𝜏), because in this case we are assuming to know the transition times. Notice that the conditioning on 𝐷 in Equation (11) also concerns the fact that (∑𝑖𝑗=1𝑡𝑛,𝑗−1→𝑗+ 𝑡𝑛,𝑖→𝑖+1) ∈ [(∑𝑖𝑗=1𝑡𝑛,𝑗−1→𝑗+
𝑡𝑛,𝑖→𝑖+1), (∑𝑖𝑗=1𝑡𝑛,𝑗−1→𝑗+ 𝑡𝑛,𝑖→𝑖+1)] ⊆ [(𝑘𝑛,𝑖+1− 1)𝜏, 𝑘𝑛,𝑖+1𝜏], if 𝛿𝑛,𝑖→𝑖+1=0.
Analogously to Equation (7), Equation (11) can be divided in three parts to divide the maximization problem into three easier sub-problems and, thus, simplify the parameter estimation problem:
ℒ(𝜗|𝐷)=ℒ1→2(𝛼1, 𝛽1|𝐷)+ℒ2→3(𝛼2, 𝛽2|𝐷)+ℒ3→4(𝛼3, 𝛽3|𝐷) (12) where ℒ𝑖→𝑖+1(𝜗𝑖|𝐷) = ∑ 𝑙𝑜𝑔 [𝑓𝑇𝑖→𝑖+1 (𝑡𝑛,𝑖→𝑖+1) 𝛿𝑛,𝑖→𝑖+1 ∙ 𝑅𝑇𝑖→𝑖+1(𝑇𝑚− ∑ 𝑡𝑛,𝑗−1→𝑗 𝑖 𝑗=1 ) (1−𝛿𝑛,𝑖→𝑖+1) ] 𝑁 𝑛=1 , 𝑖 = 1,2,3 (13)
When the information about 𝑇𝑛,𝑖→𝑖+1 is represented by the possibility distribution 𝜇𝑡̃𝑛,𝑖→𝑖+1, then the likelihood in Equation (11) reads (Denoeux, 2011):
𝐿̃(𝜗|𝐷) = ∏ [∏ 𝑓̃𝑇𝑖→𝑖+1(𝑡𝑛,𝑖→𝑖+1) 𝛿𝑛,𝑖→𝑖+1 ∙ 𝑅𝑇𝑖→𝑖+1(𝑇𝑚− ∑ 𝑡𝑛,𝑗−1→𝑗 𝑖 𝑗=1 ) (1−𝛿𝑛,𝑖→𝑖+1) 3 𝑖=1 ] 𝑁 𝑛=1 (14) where 𝑓̃𝑇𝑖→𝑖+1(𝑡𝑛,𝑖→𝑖+1) = ∫ 𝜇𝑡̃𝑛,𝑖→𝑖+1(𝑡𝑖→𝑖+1) +∞ 0 ∙ 𝑓𝑇𝑖→𝑖+1(𝑡𝑖→𝑖+1)𝑑𝑡𝑖→𝑖+1= ∫ 𝜇𝑡̃𝑛,𝑖→𝑖+1(𝑡𝑖→𝑖+1) 𝑡𝑛,𝑖→𝑖+1 𝑡𝑛,𝑖→𝑖+1 ∙ 𝑓𝑇𝑖→𝑖+1(𝑡𝑖→𝑖+1)𝑑𝑡𝑖→𝑖+1 (15)
That is, the imprecise evidence represented by the possibility distribution 𝜇𝑡̃𝑛,𝑖→𝑖+1(𝑡𝑖→𝑖+1) forces to change 𝑓𝑇𝑖→𝑖+1(𝑡𝑛,𝑖→𝑖+1) into 𝑓̃𝑇𝑖→𝑖+1(𝑡𝑛,𝑖→𝑖+1) through Equation (15), which applies the Zadeh’s definition of probability of a fuzzy event (Denoeux, 2011; Zadeh, 1996 ). Notice that 𝑅𝑇𝑖→𝑖+1 is not affected by fuzzy uncertainty as it relates to the case in which the transition has not been observed. The log-likelihood is given by:
ℒ̃(𝜗|𝐷) = log ( 𝐿̃(𝜗|𝐷)) = ℒ̃1→2(𝛼1, 𝛽1|𝐷) + ℒ̃2→3(𝛼2, 𝛽2|𝐷) + ℒ̃3→4(𝛼3, 𝛽3|𝐷) (16) where ℒ̃𝑖→𝑖+1(𝜗𝑖|𝐷)=∑ 𝑙𝑜𝑔[𝑓̃𝑇𝑖→𝑖+1(𝑡𝑛,𝑖→𝑖+1) 𝛿𝑛,𝑖→𝑖+1 ∙ 𝑅𝑇𝑖→𝑖+1(𝑇𝑚−∑𝑖𝑗=1𝑡𝑛,𝑗−1→𝑗) (1−𝛿𝑛,𝑖→𝑖+1) ] 𝑁 𝑛=1 , 𝑖 = 1,2,3 (17)
3.3. Fuzzy Expectation-Maximization algorithm
The EM algorithm has been proposed by (Dempster et al., 1977) as a broadly applicable and efficient mechanism for computing maximum likelihood estimates. It is made up of two steps (i.e., expectation and maximization), which are iteratively performed until the maximum of the likelihood function is achieved.
Inspired by the original EM algorithm, (Denoeux, 2011) has proposed an enhancement that extends its application to fuzzy evidence. We resort to this method for the maximization of each ℒ̃𝑖→𝑖+1 in
Every iteration 𝑞 of the algorithm is composed by two steps: STEP 1: Expectation step (E-step)
The expectation step requires the calculation of the expected value of the log-likelihood 𝐿̃𝑖→𝑖+1 conditional to a set of fuzzy evidences:
𝑄(𝜗𝑖, 𝜗𝑖𝑞) = 𝐸𝜗
𝑖
𝑞[ℒ̃𝑖→𝑖+1|𝐷] (18)
which reads (Denoeux, 2011):
𝑄(𝜗𝑖, 𝜗𝑖𝑞) = ∑ {∫ 𝜇𝑡̃𝑛,𝑖→𝑖+1(𝑡𝑖→𝑖+1) log[𝐿(𝜗|𝐷)] 𝐿(𝜗 𝑞|𝐷)𝑑𝑡 𝑖→𝑖+1 +∞ 0 ∫ 𝜇𝑡̃𝑛,𝑖→𝑖+1(𝑡𝑖→𝑖+1)𝐿(𝜗𝑞|𝐷)𝑑𝑡 𝑖→𝑖+1 +∞ 0 } 𝑁 𝑛=1 (19)
Then, the expected value 𝑄(𝜗𝑖, 𝜗𝑖𝑞) of ℒ̃𝑖→𝑖+1 in Equation (17) conditioned to the set of fuzzy evidences 𝐷, given the fit 𝜗𝑖𝑞 of 𝜗
𝑖 at the current iteration 𝑞, becomes:
𝑄(𝜗𝑖, 𝜗𝑖 𝑞 ) = = ∑ [∫ 𝜇𝑡̃𝑛,𝑖→𝑖+1(𝑡𝑖→𝑖+1)∙log[𝑓𝑇𝑖→𝑖+1(𝑡𝑖→𝑖+1|𝜗)]∙𝑓𝑇𝑖→𝑖+1(𝑡𝑖→𝑖+1|𝜗 𝑞)𝑑𝑡 𝑖→𝑖+1 +∞ 0 ∫0+∞𝜇𝑡̃𝑛,𝑖→𝑖+1(𝑡𝑖→𝑖+1)∙𝑓𝑇𝑖→𝑖+1(𝑡𝑖→𝑖+1|𝜗𝑞)𝑑𝑡𝑖→𝑖+1 ] 𝛿𝑛,𝑖→𝑖+1 𝑁 𝑛=1 + ∑𝑁𝑛=1log [𝑅𝑇𝑖→𝑖+1(𝑇𝑚− ∑ 𝑡𝑛,𝑗−1→𝑗 𝑖 𝑗=1 )](1−𝛿𝑛,𝑖→𝑖+1) (20)
STEP 2: Maximization step (M-step)
The maximization step consists in maximizing 𝑄(𝜗𝑖, 𝜗𝑖 𝑞
) with respect to 𝜗𝑖. To do this, we need to
select an arbitrary guess vector 𝜃𝑞=0 which here is obtained by applying the standard MLE approach
to the same dataset. The first iteration of the M-step stops when 𝜕𝑄(𝜗𝑖,𝜗𝑞 )
𝜕𝜗𝑖 is smaller than an arbitrary
fixed tolerance. The obtained parameters 𝜗𝑞=1, will be the guess vector for the following
maximization step.
The expectation and maximization steps are iterated until the difference |ℒ̃(𝜗𝑖𝑞+1) −ℒ̃(𝜗𝑖𝑞)| is smaller than some arbitrary fixed tolerance.
Once the parameters 𝛼𝑖 and 𝛽𝑖, 𝑖 = 1, 2, 3, of the HCTSMM have been estimated, they can be used to estimate the PS RUL at any 𝑘𝑡ℎ inspection time, with 𝑘 = 1, … , 𝑀
𝑛. To highlight the benefits of
including the expert knowledge in the model estimation parameters, we estimate the RUL in both the settings illustrated in Sections 3.1 and 3.2 (i.e., based only on data, and on data with additional imprecise information from the experts).
To this aim, we first consider the random variable 𝑇𝑖→𝑖+1(𝑡𝑖→𝑖+1|𝑡𝑛,𝑖0 ), which represents the residual sojourn time in state 𝑖, provided that the component has already been in state 𝑖 for 𝑡𝑛,𝑖0 years. The PDF
and CDF of 𝑇𝑖→𝑖+1 conditional on 𝑡𝑛,𝑖0 are indicated by 𝑓
𝑇𝑖→𝑖+1(𝑡𝑖→𝑖+1|𝑡𝑛,𝑖
0 ) and 𝐹
𝑇𝑖→𝑖+1(𝑡𝑖→𝑖+1|𝑡𝑛,𝑖
0 ),
respectively. Accordingly, the expected value of the random variable 𝑇𝑖→𝑖+1(𝑡𝑖→𝑖+1|𝑡𝑛,𝑖0 ),
𝐸[𝑇𝑖→𝑖+1|𝑡𝑛,𝑖0 ] is 𝐸[𝑇𝑖→𝑖+1|𝑡𝑛,𝑖0] = ∫ (𝑡𝑖→𝑖+1− 𝑡𝑛,𝑖0) +∞ 𝑡𝑛,𝑖0 𝑓𝑇𝑖→𝑖+1(𝑡𝑖→𝑖+1|𝑡𝑛,𝑖 0)𝑑𝑡 𝑖→𝑖+1= [ 𝐸[𝑇𝑖→𝑖+1] − ∫ 𝑡𝑖→𝑖+1∙ 𝑓𝑇𝑖→𝑖+1(𝑡𝑖→𝑖+1)𝑑𝑡𝑖→𝑖+1 𝑡𝑛,𝑖0 0 𝑅𝑇𝑖→𝑖+1(𝑡𝑛,𝑖0) ] − 𝑡𝑛,𝑖0 (21)
where 𝐸[𝑇𝑖→𝑖+1] represents the expected value of 𝑇𝑖→𝑖+1.
The 𝛼 −quantile 𝑞𝛼[𝑇𝑖→𝑖+1|𝑡0𝑛,𝑖] of 𝑇𝑖→𝑖+1(𝑡𝑖→𝑖+1|𝑡𝑛,𝑖0 ) can be derived from the inverse function
𝐹𝑇𝑖→𝑖+1−1(𝛼|𝑡𝑛,𝑖0 ) of 𝐹𝑇𝑖→𝑖+1(𝑡𝑖→𝑖+1|𝑡𝑛,𝑖0 ).
Suppose that at the inspection time 𝑘𝜏, the component is found in state 𝑖; then, its RUL is the random variable 𝑅𝑈𝐿(𝑘𝜏) defined as:
𝑅𝑈𝐿( 𝑘𝜏) = 𝑇𝑖→𝑖+1(𝑡𝑖→𝑖+1|𝑡𝑛,𝑖0 ) + ∑ 𝑇 𝑗→𝑗+1 𝑗>𝑖
, 𝑖, 𝑗 ∈ {1, 2, 3} (22)
assuming that the sojourn time already spent in state 𝑖 is 𝑡𝑛,𝑖0 .
The expected value, 𝐸[𝑅𝑈𝐿(𝑘𝜏)], and the 𝛼 −quantiles of 𝑅𝑈𝐿(𝑘𝜏), 𝑞𝛼[𝑅𝑈𝐿(𝑘𝜏)], are given by, respectively:
𝐸[𝑅𝑈𝐿(𝑘𝜏)] = { {𝐸[𝑇1→2] − ∫ 𝑡1→2∙ 𝑓𝑇1→2(𝑡1→2)𝑑𝑡1→2 𝑡𝑛,𝑖0 0 𝑅𝑇1→2(𝑡𝑛,𝑖 0) } + 𝐸[𝑇2→3] + 𝐸[𝑇3→4] − 𝑘𝜏, 𝑖𝑓 𝑖 = 1 {𝐸[𝑇2→3] − ∫ 𝑡2→3∙ 𝑓𝑇2→3(𝑡2→3)𝑑𝑡2→3 𝑡𝑛,𝑖0 0 𝑅𝑇2→3(𝑡𝑛,𝑖 0) } + 𝐸[𝑇3→4] − 𝑡𝑛,𝑖0 , 𝑖𝑓 𝑖 = 2 {𝐸[𝑇3→4] − ∫ 𝑡3→4∙ 𝑓𝑇3→4(𝑡3→4)𝑑𝑡3→4 𝑡𝑛,𝑖0 0 𝑅𝑇3→4(𝑡𝑛,𝑖 0 ) } − 𝑡𝑛,𝑖0 , 𝑖𝑓 𝑖 = 3 (23) 𝑞𝛼[𝑅𝑈𝐿(𝑘𝜏)]: ∫ 𝑓𝑅𝑈𝐿(𝑘𝜏)(𝑡)𝑑𝑡 = 𝛼 𝑞𝛼[𝑅𝑈𝐿(𝑘𝜏)] 0 (24)
where 𝑓𝑅𝑈𝐿(𝑘𝜏) is the PDF of the random variable 𝑅𝑈𝐿( 𝑘𝜏).
The definition of 𝑡𝑛,𝑖0 depends on the case under investigation. In details, in the setting described in
Section 3.1 with data only, we know that 𝑡𝑛,10 = 𝑘𝜏 and 𝑡
𝑛,𝑖0 𝜖 [(𝑘𝜏 − 𝑘𝑛,𝑖𝜏), (𝑘𝜏 − (𝑘𝑛,𝑖− 1)𝜏)], 𝑖 =
2,3. In this case, 𝐸[𝑅𝑈𝐿(𝑘𝜏)] and 𝑞𝛼[𝑅𝑈𝐿(𝑘𝜏)] are interval estimates, whose lower and upper
bounds are obtained by substituting 𝑡𝑛,𝑖0 with (𝑘𝜏 − 𝑘
𝑛,𝑖𝜏) and (𝑘𝜏 − (𝑘𝑛,𝑖− 1)𝜏) (see Figure 3),
respectively, in Equations (23) and (24).
Figure 3: 𝒕𝒏,𝒊𝟎 for setting in Section 3.1.
With respect to the setting illustrated in Section 3.2, the sojourn time in state 𝑖 corresponds to the difference between the crisp number (𝑘𝜏 − 𝑘𝑛,𝑖−1𝜏) and the triangular fuzzy number 𝑡̃𝑛,𝑖−1→𝑖, resulting in the possibility distribution 𝜇𝑡̃
𝑛,𝑖0 (𝑡𝑛,𝑖
0 ) (Aven et al., 2014):
𝜇𝑡̃
𝑛,𝑖
which is shown in Figure 4.
Figure 4: 𝒕𝒏,𝒊𝟎 for setting in Section 3.2.
Since the random variable 𝑅𝑈𝐿(𝑘𝜏) depends on the sojourn time, as shown in Equation (22), it is affected by the epistemic uncertainty on 𝑡𝑛𝑖0, and, thus, in the case considered it becomes a fuzzy
number, described by a possibility distribution. Therefore, the expected value and the 𝛼 −quantile estimates of the fuzzy 𝑅𝑈𝐿(𝑘𝜏), according to the Zadeh extension principle, are fuzzy numbers too (Zadeh, 1996; Aven et al., 2014). The lower bound, core and upper bound of the estimates of 𝐸[𝑅𝑈𝐿(𝑘𝜏)] and 𝑞𝛼[𝑅𝑈𝐿(𝑘𝜏)] are obtained by combining Equation (25) with Equation (23) and Equation (24), respectively.
Finally, notice that if at inspection time 𝑘𝜏 the component is found in state 1 (i.e., no transition has been observed), then 𝐸[𝑅𝑈𝐿(𝑘𝜏)] and 𝑞𝛼[𝑅𝑈𝐿(𝑘𝜏)] do not depend on 𝑡𝑛,𝑖0 .
5. CASE STUDY
In this Section, the settings described in Sections 3.1 and 3.2 and the RUL estimation procedure discussed in Section 4 are applied to a simulated case study concerning the 4-states PS degradation process described in Section 2.
5.1. Data Simulation
We have artificially generated N = 100 degradation paths by Monte-Carlo (MC) sampling from the Markov Model described in (Fleming, 2004), which assumes that the transition times are exponentially distributed with scale parameters 𝜆𝑖→𝑖+1 given in Table 1.
𝑖 𝜆𝑖→𝑖+1(𝑦−1)
1 4.35×10−4
3 1.97×10−2
Table 1: Values of the scale parameter 𝝀𝒊→𝒊+𝟏 used to simulate the degradation process (Fleming, 2004).
Then, a right censoring mechanism has been applied to the gathered data, with 𝑇𝑚 = 60 𝑦𝑒𝑎𝑟𝑠. This value is taken from (Di Maio et al., 2015) and considers that typically NPPs have a life time of 40 years, plus possible extension. We consider that the system is periodically inspected with period 𝜏 = 5 𝑦𝑒𝑎𝑟𝑠.
The case study dataset is summarized in Table 2a, which reports for every state 𝑖 (first column) the corresponding number of components that have entered in state 𝑖 (second column) and the number of components which are found in state 𝑖 at inspection 𝑘𝑛,𝑖= 𝑀𝑛 (third column).
Table 2a: Case study
dataset.
To better understand the case study dataset, the degradation paths of the six components that entered a state 𝑖 > 1 are summarized in Table 2b, where the first column reports the 𝑛𝑡ℎ component, whereas the other three columns report the values of 𝑘𝑛,𝑖, 𝑖 = 2,3,4, respectively, as defined in Subsection 2.1.
Table 2b: Case study dataset.
𝑖 Number of components entered in state 𝑖 Number of components found in state 𝑖 at inspection 𝑀𝑛 1 100 94 2 6 4 3 2 1 4 1 1 𝑛 𝑘𝑛, 2 𝑘𝑛,3 𝑘𝑛,4 4 4 - - 13 11 - - 17 11 - - 41 8 9 - 74 3 - - 91 1 10 12
Obviously, in this simulated case study we do not have real expert judgments. Nevertheless, for a better understanding of how the expert knowledge can influence the estimation of the unknown parameters, we investigate three different settings relating to as many risk attitudes of the expert. Namely, we consider three different types of possibility distributions 𝜇𝑡̃𝑛,𝑖→𝑖+1(𝑡𝑛,𝑖→𝑖+1)
corresponding to the information about the transition time retrievable from a moderately risk-averse (Figure 5), risk- averse (Figure 6), and risk- prone (Figure 7) expert. For brevity, these three settings will be referred to as cases 𝐶1, 𝐶2 and 𝐶3, respectively, whereas the MLE setting will be referred to as 𝐶0.
Figure 5: Example of possibility distribution of case 𝑪𝟏 (moderately risk-averse).
Figure 7: Example of possibility distribution of case 𝑪𝟑 (risk-prone).
In the first case, the expert does not commit her/him-self and, thus, does not reduce the support of the possibility distribution; she/he only gives the fully plausible value of 𝑡𝑛,𝑖→𝑖+1, within the interval
[(𝑘𝑛,𝑖+1− 1)𝜏; 𝑘𝑛,𝑖+1𝜏], but closer to its lower bound; namely, he/she sets 𝑡⏞𝑛,𝑖→𝑖+1 = 0.667 + 𝑡̇𝑛,𝑖0 ,
where 𝑡̇𝑛,𝑖0 = (𝑘
𝑛,𝑖+1− 1)𝜏 − 𝑘𝑛,𝑖𝜏 (i.e., the time elapsed from the first inspection time in which the
component has been found in state 𝑖 and the last one). In the second case, the expert is more conservative, i.e., he/she feels that the transition surely lies in the interval [(𝑘𝑛,𝑖+1− 1)𝜏; (𝑘𝑛,𝑖+1𝜏 − 2 )], with core 𝑡⏞𝑛,𝑖→𝑖+1 = 0.667 + 𝑡̇𝑛,𝑖0 . Finally, the case in Figure 7 refers to an expert that feels
confident on the system and, thus, states that the transition surely occurred in the interval
[(𝑘𝑛,𝑖+1− 1)𝜏 + 2); 𝑘𝑛,𝑖+1𝜏 ], the fully plausible value 𝑡⏞𝑛,𝑖→𝑖+1 = 2.667 + 𝑡̇𝑛,𝑖0 .
5.2. Parameters Estimation results
Tables 3-6 report the results obtained for cases 𝐶0− 𝐶3, respectively, which are summarized by the
expected value, 𝐸[𝑇𝑖→𝑖+1], the variance, 𝑉𝑎𝑟[𝑇𝑖→𝑖+1], the median, 𝑞0.5[𝑇𝑖→𝑖+1] and the interval 𝐼 =
[𝑞0.05[𝑇𝑖→𝑖+1]; 𝑞0.95[𝑇𝑖→𝑖+1]] which covers 90 % of the values of the random variables 𝑇𝑖→𝑖+1 , 𝑖 =
1,2,3 , obeying Weibull distributions.
MLE estimation
1 𝛼1=1.494e+03 𝛽1=0.866 3. 466e+06 1.607e+03 979.22 [50.15; 5319.51] 2 𝛼2=104.41 𝛽2=0.65 5.097e+04 142.13 59.52 [1.07; 561.88] 3 𝛼3=21.29 𝛽3=1.44 185.04 19.32 16.52 [2.71; 45.62]
Table 3: Results for case 𝑪𝟎.
FEM estimation 𝒊 𝛝̂𝐅𝐄𝐌 𝑽𝒂𝒓{𝑻𝒊→𝒊+𝟏} 𝑬{𝑻𝒊→𝒊+𝟏} 𝒒𝟎.𝟓{𝑻𝒊→𝒊+𝟏} I 1 𝛼1=1.671e+03 𝛽1=0.837 4.858e+07 1.835e+03 1078.75 [47.48; 6239.39] 2 𝛼2=98.26 𝛽2=0.68 3.698e+04 127.60 57.41 [1.28; 484.85] 3 𝛼3=21.57 𝛽3=1.32 229.89 19.85 16.34 [2.23; 49.47]
Table 4: Results for case 𝑪𝟏.
The parameters values estimated in case 𝐶1 are similar to those in case 𝐶0 . This is due to the fact that
the simulated dataset is composed by 94% of components that have never experienced any transition during the mission time i.e., they are still in state 1 at 𝑡 = 𝑇𝑚 (Table 2). For these components, the expert opinion on state transition times is not exploited. Moreover, in case 𝐶1, the expert gives a possibility distribution whose support is coincident with the interval between successive inspections, which corresponds to the interval-censored data considered in case 𝐶0.
FEM estimation 𝒊 𝛝̂𝐅𝐄𝐌 𝑽𝒂𝒓{𝑻𝒊→𝒊+𝟏} 𝑬{𝑻𝒊→𝒊+𝟏} 𝒒𝟎.𝟓{𝑻𝒊→𝒊+𝟏} I 1 𝛼1=2.134e+03 𝛽1=0.779 1.020e+06 2.464e+03 1334.30 [47.71; 8753.99]
2 𝛼2=108.69 𝛽2=0.61 7.018e+04 156.91 60.18 [0.92; 640.69] 3 𝛼3=21.90 𝛽3=1.20 293.80 20.57 16.16 [1.87; 54.36]
Table 5: Results for case 𝑪𝟐.
FEM estimation 𝒊 𝛝̂𝐅𝐄𝐌 𝑽𝒂𝒓{𝑻𝒊→𝒊+𝟏} 𝑬{𝑻𝒊→𝒊+𝟏} 𝒒𝟎.𝟓{𝑻𝒊→𝒊+𝟏} I 1 𝛼1=1.011e+03 𝛽1=0.985 1.066e+07 1.017e+03 697.54 [50.66; 3085.01] 2 𝛼2=77.92 𝛽2=0.89 8.427e+03 82.16 51.78 [2.86; 265.42] 3 𝛼3=20.73 𝛽3=1.64 133.69 18.54 16.59 [3.38; 40.21]
Table 6: Results for case 𝑪𝟑.
The impact of including the expert information is appreciable when case 𝐶0 is compared with cases 𝐶2 and 𝐶3, where the supports of the possibility distributions on the transition times do not coincide with
the whole interval [(𝑘𝑛,𝑖+1− 1)𝜏; 𝑘𝑛,𝑖+1𝜏]. We consider only the transitions observed before 𝑇𝑚, so that the parameter estimation is influenced by transition times 𝑡𝑛,𝑖→𝑖+1 < 60 for which the expert expresses his/her opinion. The influence of the expert risk-attitude can be appreciated when comparing the estimated parameters 𝛽. In particular, for the first and the second transition, 𝛽1 and 𝛽2 are smaller than 1, which means decreasing estimated transition rate over time, but, in case 𝐶3, 𝛽1 and 𝛽2 are larger than the estimates of cases 𝐶0 and 𝐶2.
In order to clarify the influence of the expert risk-attitude on transition rates, we have computed 𝜆𝐶0 for case 𝐶0 (i.e., the expert is moderately risk-averse), 𝜆𝐶2 for case 𝐶2 (i.e., the expert is risk-averse), and 𝜆𝐶3 for case 𝐶3 (i.e, the expert is moderately risk-prone), according to the expression for 𝜆𝑖 given
in Equation (4). The behaviors over time of the estimated transition rates for the transition from state 1 to state 2 are shown in Figure 8: in the initial part of the time axis, the transition rate related to the risk-prone expert is smaller than those of the moderately risk-averse and risk-averse experts.
Figure 8: Transition rates for transition from state 1 to state 2, with 𝝀𝑪𝟎 for Case 𝑪𝟎, 𝝀𝑪𝟐 for Case 𝑪𝟐 and 𝝀𝑪𝟑 for
Case 𝑪𝟑.
Similarly, the rate for the third transition estimated in case 𝐶3 is initially larger than those in case 𝐶0 and 𝐶2, as shown in Figure 9.
Figure 10 shows the estimated reliability function for the different scenarios, which has been derived by the Monte Carlo (MC) approach summarized in Appendix A.
Figure 10: Estimated reliability function.
5.3.
RUL estimation results
In this Section, we report the results of the estimation of 𝐸 [𝑅𝑈𝐿(𝑘𝜏)] and 𝑞𝛼[𝑅𝑈𝐿(𝑘𝜏)], by applying
the procedure described in Section 4.
5.3.1 𝐸[𝑅𝑈𝐿(kτ)] estimation
The expected value 𝐸 [𝑅𝑈𝐿((𝑘𝜏))] of the random variable 𝑅𝑈𝐿(𝑘𝜏) has been estimated according to Equation (23) at each inspection time 𝑘𝜏, 𝑘 = 1 … 𝑀𝑛, assuming that at this time instant the component can be found in state 1, state 2 or state 3.
Figures 11-16 show the evolution of 𝐸 [𝑅𝑈𝐿(𝑘𝜏)] over 𝑘𝜏. We firstly compare the results obtained in case 𝐶0 with those obtained in case 𝐶1, as their estimated model parameters are similar to each
other. Then, RUL estimates in case 𝐶0 are compared to those of cases 𝐶2 and 𝐶3. For visualization, the axes of these Figures have different scales; yet, for states 𝑖 = 2, 3 the 𝐸 [𝑅𝑈𝐿(𝑘𝜏)] is divided into two plots: a) where 𝑘𝜏 = {5, … ,30} and b) where 𝑘𝜏 = {35, … ,60}. The estimates of 𝐸 [𝑅𝑈𝐿(𝑘𝜏)], given that the PS is in state 1, are reported in Figure 11, both for case 𝐶1 and for case 𝐶0. Since no transition has been observed when the component is found in state 1, then the estimates do not depend on 𝑡𝑛,𝑖0 and are not affected by uncertainty. The difference between the estimated RULs is only due
Figure 11: Results for Cases 𝑪𝟎 and 𝑪𝟏; state 1; E[RUL] and Inspection time in years.
In Figures 12-13, the estimates of 𝐸 [𝑅𝑈𝐿(𝑘𝜏)] for states 2 and 3 are shown, respectively, which are intervals for case 𝐶0 and triangular possibility distributions for case 𝐶1. From the comparison of these Figures, we can notice that in case 𝐶1 the uncertainty on 𝐸 [𝑅𝑈𝐿((𝑘𝜏))] is always smaller than that
in case 𝐶0, at any inspection time. This is due to the fact that the expert statements introduce an additional information on the system behaviour, which allows better specifying the value of 𝐸 [𝑅𝑈𝐿(𝑘𝜏)] in the interval containing its unknown value. This result highlights the contribution of the method here proposed to exploit any source of information that can corroborate data.
Figure 13: Results for cases 𝑪𝟎 and 𝑪𝟏; state 3; E[RUL] and Inspection time in years.
With respect to the comparison of case 𝐶0 to cases 𝐶2 and 𝐶3, Figure 14 compares the 𝐸 [𝑅𝑈𝐿(𝑘𝜏)] estimates when the component is in state 1, which do not depend on 𝑡𝑛,𝑖0 . The estimates are not
affected by uncertainty, as introduced previously, and the ones of case 𝐶0 are always smaller than
those of case 𝐶2 and larger than those of case 𝐶3. This difference reflects the different expected value estimated in each case, in fact the expected value of case 𝐶2 is larger than the others, while the one of case 𝐶3 is smaller.
Figure 14: Results for cases 𝑪𝟎, 𝑪𝟐 and 𝑪𝟑; state 1; E[RUL] and Inspection time in years.
Figure 15 compares the results of case 𝐶0 to those of cases 𝐶2 and 𝐶3, when the PS is found in state 2. We can note that intervals estimated in case 𝐶0 never contain the fuzzy estimates of cases 𝐶2 and 𝐶3.
Figure 15: Results for cases 𝑪𝟎, 𝑪𝟐 and 𝑪𝟑; state 2; E[RUL] and Inspection time in years.
In particular, the estimates in case 𝐶3 are always smaller than the corresponding ones of case 𝐶0,
which are smaller than those of case 𝐶2. This is due to the fact that the estimates of 𝐸[𝑇𝑖→𝑖+1], 𝑖 = 1, 2, 3, are larger in case 𝐶2 than in case 𝐶0, while they are larger in case 𝐶0 than in case 𝐶3. Figure 16 shows the RUL estimates in cases 𝐶0, 𝐶2 and 𝐶3 assuming that the component is found in state 3.
In this case, the estimates are closer to each other since the corresponding estimated parameters 𝜗 for the third transition are similar to each other, as underlined previously.
Figure 16: Results for cases 𝑪𝟎, 𝑪𝟐 and 𝑪𝟑; state 3; E[RUL] and Inspection time in years.
5.3.2 𝒒𝜶[𝑹𝑼𝑳(𝒌𝝉)]estimation
“The interval containing 𝛼% of the values of the random variable 𝑅𝑈𝐿(𝑘𝜏), Equation (24), has been estimated through MC simulation (see Appendix B), at each inspection time 𝑘𝜏, 𝑘 = 1 … 𝑀𝑛,
assuming that at this time instant the component can be found in state 1, state 2 or state 3.
Figures 17-22 show the evolution 𝑞𝛼[𝑅𝑈𝐿(𝑘𝜏)] over 𝑘𝜏 and we report the estimates for 𝛼 = 0.1. We firstly compare the results obtained in case 𝐶0 with those obtained in case 𝐶1, as their estimated model parameters are similar to each other. Then RUL estimates in Case 𝐶0 are compared to those of cases
𝐶2 and 𝐶3 . For visualization, the axes of these Figures have different scales; yet, for states 𝑖 = 2,3 the 𝑞𝛼[𝑅𝑈𝐿(𝑘𝜏)] is divided into two plots: a) where 𝑘𝜏 = {5, … ,30} and b) where 𝑘𝜏 = {35, … ,60}. The estimates of 𝑞𝛼[𝑅𝑈𝐿(𝑘𝜏)] , given that the PS is in state 1, are reported in Figure 15, both for case
𝐶1 and for case 𝐶0. Since no transition has been observed when the component is found in state 1, then the estimates do not depend on 𝑡𝑛,𝑖0 and are not affected by uncertainty.
Figure 17: Results for case 𝑪𝟎 , 𝑪𝟏, state 1; 𝒒𝟎.𝟏[𝑹𝑼𝑳] and Inspection time in years.
In Figures 18-19, the estimates are shown for the states 2 and 3, respectively, which are intervals for case 𝐶0 and traingular possibility distributions for case 𝐶1. From the comparison of these Figures, we can notice that in case 𝐶1 the uncertainty on 𝑞𝛼[𝑅𝑈𝐿(𝑘𝜏)] is always smaller than that in case 𝐶0, at any inspection time. This is thanks to the additional information given by the expert and exploited in the estimation, which allows better specifying the value of 𝑞𝛼[𝑅𝑈𝐿(𝑘𝜏)] in the interval containing
its unknown value.
Figure 18: Results for case 𝑪𝟎 , 𝑪𝟏, state 2; 𝒒𝟎.𝟏[𝑹𝑼𝑳] and Inspection time in years.
With respect to the comparison of case 𝐶0 to cases 𝐶2 and 𝐶3, Figure 20 compares the 𝑞𝛼[𝑅𝑈𝐿(𝑘𝜏)] estimates when the component is in state 1. The estimates of case 𝐶2 are larger than the others, throughout the time range, reflecting the larger expected values estimated for the transition times, in this case. Whereas, the estimates of case 𝐶3 are smaller than the others, over all the time range,
reflecting the smaller expected values estimated for the transition times, in this case.
Figure 20: Results for case 𝑪𝟎 , 𝑪𝟐 and 𝑪𝟑 state 1; 𝒒𝟎.𝟏[𝑹𝑼𝑳] and Inspection time in years.
Figures 21 compares the results of case 𝐶0 to those of cases 𝐶2 and 𝐶3, in case the PS is found in state 2. We can note that, initially, the intervals estimated in case 𝐶0 almost contain the fuzzy estimates of
case 𝐶2 and are very close to those of case 𝐶3. Then, after 𝑘𝜏 = 25 years, the estimates of case 𝐶3 are smaller and differ from the others. This is due to the fact that initially the influence of the expert opinion is more relevant: in fact the estimates of case 𝐶3, in which the expert is risk-prone, are larger than those of cases 𝐶0 and 𝐶2. Figure 22 shows the RUL estimates in cases 𝐶0 and 𝐶3, assuming that
the component is found in state 3. In this case, the estimates are closer to each other and sometimes the intervals estimates of case 𝐶0 contain the possibility distributions of cases 𝐶2 and 𝐶3. As for Figure 21, initially the estimates of case 𝐶3 are larger than those of cases 𝐶2 and 𝐶0, according to the expert opinion. Furthermore, the estimates of cases 𝐶2 and 𝐶3 are close to each other since the behavior
Figure 21: Results for case 𝑪𝟎 , 𝑪𝟐 and 𝑪𝟑, state 2; 𝒒𝟎.𝟏[𝑹𝑼𝑳] and Inspection time in years.
Figure 22: Results for case 𝑪𝟎 , 𝑪𝟐 and 𝑪𝟑, state 3; 𝒒𝟎.𝟏[𝑹𝑼𝑳] and Inspection time in years.
Overall, for all the cases considered, the estimated 𝐸[𝑅𝑈𝐿(𝑘𝜏)] and 𝑞𝛼[𝑅𝑈𝐿(𝑘𝜏)] increase
proportionally to 𝑘𝜏 when the component is in state 1 or 2, independently on the case considered, and decrease when the component is in state 3. This is due to the fact that for 𝑖 = 1,2 𝛽𝑖 is less than 1, which corresponds to a decreasing failure rate (Thoman et al., 1969). Consequently, for increasing value of 𝑘𝜏, 𝐸[𝑅𝑈𝐿(𝑘𝜏)] and 𝑞𝛼[𝑅𝑈𝐿(𝑘𝜏)] are increasing over time. On the other hand, for state 𝑖 =
3, 𝛽3 is larger than 1, which corresponds to an increasing failure rate, and so, for increasing values of 𝑘𝜏, the expected value 𝐸[𝑅𝑈𝐿(𝑘𝜏)] and the quantile 𝑞𝛼[𝑅𝑈𝐿(𝑘𝜏)] are decreasing over time.
6. Conclusions
In this work, we have developed a method based on the FEM algorithm to estimate the parameters of a MS degradation model. The method allows integrating field data from inspection outcomes with additional information about the state transition times from maintenance operators. Such additional, imprecise information has been represented by possibility distributions. Based on the MS model with estimated parameters, a procedure for predicting the RUL has been developed. The proposed method has been applied to a case study concerning the degradation of pipe welds in the coolant system of a PWR NPP. We have also investigated how results change when the expert knowledge is not employed and only inspection outcomes are considered. The results have shown that the combination of field data with expert knowledge allows reducing the uncertainty in degradation estimation. Finally, the proposed methodology can be easily extended to other industrial reliability problems, where information from expert is available to supplement field data.
Acknowledgements
The participation of Enrico Zio to this research is partially supported by the China NSFC under grant number 71231001.
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